- Open Access
The upper bound estimate of the number of integer points on elliptic curves
© Zhang and Li; licensee Springer. 2014
- Received: 22 January 2014
- Accepted: 20 February 2014
- Published: 4 March 2014
Let p be a fixed prime and r be a fixed positive integer. Further let denote the number of pairs of integer points on the elliptic curve with . Using some properties of Diophantine equations, we give a sharper upper bound estimate for . That is, we prove that , except with , where s is a nonnegative integer.
- elliptic curve
- integer point
- Diophantine equation
where r is a positive integer.
Using some properties of Diophantine equations, we give a sharper upper bound estimate for , the number of pairs of non-trivial integer points on (1.2). That is, we shall prove the following results.
, , , where m, s are nonnegative integers.
- (ii), , , where s is a nonnegative integer, is a solution of the equation(1.5)
Theorem 1.2 Let p be an odd prime, r be a positive integer. Then for any nonnegative integer s, we have , except with . Moreover, if , then , except with .
Lemma 2.1 ([, Theorem 244])
can be expressed as , where n is a positive integer.
Lemma 2.2 If , then , where m is a nonnegative integer.
Proof Assume that n has an odd divisor d with . Then we have either and or and . Therefore, since p is a prime, it is impossible. Thus, we get . The lemma is proved. □
Any fixed positive integer a can be uniquely expressed as , where b, c are positive integers with b is square free. Then b is called the quadratfrei of a and denoted by .
Lemma 2.3 For any positive integer m, we have .
Since , where is the Legendre symbol, we see from (2.3) that for . Therefore, since , by (2.2), we obtain . It implies that . The lemma is proved. □
Then () are all solutions of (2.4).
Lemma 2.4 ([, Theorem 1])
has at most one solution . Moreover, if the solution exists, then , where .
Lemma 2.5 ([, Theorem 3])
If , then (2.5) has no solutions .
Lemma 2.6 If , where m is a positive integer with , then (1.5) has no solutions .
has solution and its fundamental solution is . Further, since , by Lemma 2.3, we have . Hence, we get . Therefore, by Lemma 2.5, the lemma is proved. □
Lemma 2.7 ()
has no solutions .
has only the solutions , where m is a nonnegative integer.
Proof Assume that is a solution of (2.9). If , since , then we have , , , and . But since is not a square, it is impossible.
Further, by Lemma 2.2, we see from the first equality of (2.13) that . Thus, by (2.10) and (2.13), the lemma is proved. □
Therefore, by (3.1), (3.4), and (3.6), the integer points of type (i) are given.
It implies that is a solution of (1.5). Therefore, by (3.1) and (3.8), we obtain the integer points of type (ii).
Since and , we see from (3.10) that is a square, a contradiction.
To sum up, the theorem is proved.
It implies that the theorem is true for .
and . However, by Lemma 2.6, if with , then . Therefore, by (4.2), if , then , except with . The theorem is proved.
The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the P.E.D. (2013JK0573) and N.S.F. (11371291) of P.R. China.
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